Asymptotic freeness with little randomness and the Weil representation
of SL_2(F_p)

We try to make the case that the Weil (a.k.a. oscillator)
representation of SL_2(F_p) could be a good source of interesting (not-very-)random
matrix problems.We do so by proving some asymptotic freeness results
and suggesting problems for research.

In particular, we offer an answer (by no means definitive) to a question
posed in a recent paper of Anderson and Farrell. We assume no familiarity
on the part of the audience with the Weil representation and will
explain how to construct it in down-to-earth and explicit fashion.

Basu,
Madhushree
Institute of Mathematical Sciences

Continuous Courant-Fischer-Weyl minimax theorem

A minimax theorem is a result that gives a characterization
of eigenvalues of compact self-adjoint operators on Hilbert spaces.
The Courant-Fischer-Weyl minmax theorem gives an extremum property of
the ${k^{th}}$ eigenvalue of a Hermitian $n \times n$ scalar matrix
(${1 \le k \le n}$), without referring to any eigenvector. Our work
grew out of the search for an extension of this theorem to a finite
von Neumann algebraic setting. We prove a version of this theorem for
a self-adjoint element having a non atomic distribution in a ${II_1}$
factor, and we also indicate an alternate proof for the finite dimensional
version of the original theorem. This noncommutative analogue of the
CFW minmax theorem uses the distribution function of the self-adjoint
operator as its main tool and makes use of a continuous version of Ky
Fan's theorem, which we state but do not prove in this talk. We finally
briefly discuss an application of the CFW theorem.
This is a joint work with V. S. Sunder.

Belinschi,
SerbanQueen's University

Spectral and Brown measures of polynomials in free random variables

The combination of a selfadjoint linearization trick due to
Greg Anderson with Voiculescu's subordination for operator-valued free
convolutions and analytic mapping theory turns out to provide a method
for finding the distribution of any selfadjoint polynomial in free variables.
In this talk we will present the analytic machinery behind this process,
and show an extension that allows in principle the computation of Brown
measures of possibly non-selfadjoint polynomials in free variables.
We will also indicate some results on Brown measures of some sums and
products of free random variables. This talk is based on joint work
with Tobias Mai and Roland Speicher and ongoing joint work with Piotr
Sniady and Roland Speicher.

Biane,
Philippe
Universite Paris-Est

From noncrossing partitions to ASM

Noncrossing partitions are related to alternating sign matrices
through the Razumov-Stroganov (ex)-conjecture. I will review this as
well as attempts to relate ASMs with some plane partitions.

We consider large Information-Plus-Noise type matrices of
the form ${M_N =(\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac {X_N}{\sqrt{N}}+A_N)^*}$
where ${X_N}$ is an ${n \times N (n \leq N)}$ matrix consisting of independent
standardized complex entries, ${A_N}$ is an ${n \times N}$ nonrandom
matrix and ${\sigma > 0}$. As N tends to infinity, if ${c_N = n/N
\rightarrow c \in [0, 1]}$ and if the empirical spectral measure ${\mu_{A_N{A_N}^*}}$
of ${A_N{A_N}^*}$ converges weakly to some compactly supported probability
distribution ${\nu \neq \delta_0}$, Dozier and Silverstein established
that almost surely the empirical spectral measure of ${M_N}$ converges
weakly towards a nonrandom distribution ${\mu_{\sigma,\nu,c}}$. Bai
and Silverstein proved, under certain assumptions on the model, that
for some fixed closed interval in ${]0;+\infty[}$ outside the support
of ${\mu_{\sigma,\mu_{A_N{A_N}^*},c_N}}$ for all large N, almost surely,
no eigenvalues of ${M_N}$ will appear in this interval for all N large.
We show that there is an exact separation phenomenon between the spectrum
of ${M_N}$ and the spectrum of ${A_N{A_N}^*}$: to a gap in the spectrum
of ${M_N}$ pointed out by Bai and Silverstein, it corresponds a gap
in the spectrum of ${A_N{A_N}^*}$ which splits the spectrum of ${A_N{A_N}^*}$
exactly as that of ${M_N}$. We deduce a relationship between the distribution
functions of some probability measures on ${\mathbb{R^+}}$ and their
rectangular free convolution with ratio c with the pushfoward of a Marchenko-Pastur
distribution with parameter c by x${\mapsto \sqrt{x}}$. We use the previous
results to characterize the outliers of spiked Information-Plus-Noise
type models.

The difference between the non-commutative and the commutative
moments of ABAB factor. This nifty little fact extends to the finite
dimensional case of random matrix theory allowing for a fourth moment
interpolation between free and classical probability that is suitable
for applications. We describe an application to a problem in quantum
many body physics, and mention comparisons with other interpolations
between free and classical probability. The method of ghosts and shadows
will be used and briefly discussed.
This is joint work with Ramis Movassagh.

Farrell,
Brendan
California Institute of Technology

Structured Random Unitary Matrices and Asymptotic Freeness

A fundamental theorem of Voiculescu relating free probability
and random matrix theory states that conjugating deterministic matrices
by Haar-distributed unitary matrices yields asymptotic freeness. In
work with Greg Anderson we show the existence of random unitary matrices
having more structure and less randomness yet also yielding asymptotic
freeness. We discuss how this work relates to discrete uncertainty principles
and classical random matrix theory.

Fevrier,
Maxime
Université Paris-Sud 11

Outliers in the Spectrum of Spiked Deformations of Unitarily Invariant
Random Matrices

We investigate the asymptotic behavior of the eigenvalues
of the random matrix A+U*BU, where A and B are deterministic Hermitian
matrices and U is drawn from the unitary group according to Haar measure.
We discuss the existence and localization of "outliers", i.e.
eigenvalues lying outside from the bulk of the spectrum. This is joint
work with S. Belinschi, H. Bercovici and M. Capitaine.

Guionnet,
Alice Massachusetts Institute of Technology

Heavy tails random matrices

We will discuss properties of the sepctrum and eigenvectors
of random matrices with possibly large entries, such as the covariance
matrix of Erdos Renyi graphs or random matrices with alpha-stable entries.
This talk is based on joint works with Bordenave, Benaych-Georges and
Male.

Koestler,
Claus
University College Cork

Quantum symmetric states on free product C*-algebras

Recently Roland Speicher and I had found a characterization
of freeness with amalgamation by quantum exchangeable random variables
in a W*-algebraic setting of probability spaces. In this talk we introduce
quantum symmetric states in a C*-algebraic setting of probability spaces
which extends the notion of quantum exchangeable random variables. Our
main result is a de Finetti type theorem for quantum symmetric states
and a characterization of extreme quantum symmetric states. We will
give some examples and, in particular, show that central quantum symmetric
states form a Choquet simplex whose extreme points are free product
states. Roughly speaking our results provide the free probability counterpart
of Stoermer's work on symmetric states on the infinite minimal tensor
product of a unital C*-algebra. This is joint work with Ken Dykema and
John Williams.

Lenczewski,
Romuald
Wroclaw University of Technology

Matricial freeness and random matrices

I will discuss the concept of matricial freeness and its
applications to the study of limit distributions of independent random
matrices. In particular, I will show how to construct a random matrix
model for free Meixner laws and the associated ensemble.

The talk is based on joint work with A. Nica. I will describe
a correspondence between free multiplicative convolution of distributions
on a noncommutative probability space and convolution of characters
in the Hopf algebra of symmetric functions. The correspondence is established
via twisted parking symmetric functions. I will explain how these symmetric
functions can also arise from representation theory and mention some
connections with diagonal harmonics.

Mingo,
James Queen's University at Kingston

Asymptotic Freeness of Orthogonally and Unitarily Invariant Ensembles

There has been a strong relation between unitary invariance
and asymptotic freeness ever since Voiculescu's 1991 paper on asymptotic
freeness. At the first order level there is very little difference between
the case of orthogonally and unitarily invariant ensembles. Above this
level the transpose plays a significant role in the orthogonal case,
something which isn't seen in the unitary case. This means one has to
consider ensembles ${\{A_N\}}$ in which there is joint limit distribution
for words in ${A_N}$ and ${A_N^t}$, i.e. a limit ${t}$-distribution.
When one has an ensemble which has a limit ${t}$-distribution and is
also unitarily invariant one gets the surprising result that ${A_N}$
and ${A_N^t}$ become free. In particular this applies to ensembles of
Haar distributed random unitary operators.

This is joint work with Mihai Popa.

Ramis
Movassagh
Northeastern University

Isotropic Entanglement

The method of "Isotropic Entanglement" (IE), inspired
by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue
distribution of quantum many-body (spin) systems with generic interactions.
At the heart is a "Slider", which interpolates between two
extrema by matching fourth moments. The first extreme treats the non-commuting
terms classically and the second treats them isotropically. Isotropic
means that the eigenvectors are in generic positions. We prove Matching
Three Moments and Slider Theorems and further prove that the interpolation
is universal, i.e., independent of the choice of local terms. Our examples
show that IE provides an accurate picture well beyond what one expects
from the first four moments alone.

Muraki,
Naofumi
Iwate Prefectural University

On a q-deformation of free independence

I will introduce a product operation (q-product) for non-commutative
probability spaces. There exists naturally a notion of `q-independence'
which is associated to the q-product. Here an independence is understood
as a universal calculation rule for mixed moments of non-commutative
random variables, in the sense of Speicher but this time we drop the
associativity condition for such rule. The existence of universal calculation
rule for q-product can be proved based on the card arrangement. The
q-deformed cumulants is also discussed.

Nica, Alexandru
University of Waterloo

Some remarks on the combinatorics of free unitary Brownian motion

It is well-known that, if ${u}$ is a Haar unitary, then the
joint free cumulants of ${u}$ and ${u^{*}}$ have a very nice explicit
description. The situation is not at all the same when instead of ${u}$
we look at a unitary ${u_t}$ (with ${t}$ in ${[0, \infty )}$) from the
free unitary Brownian motion. In my talk I will present some remarks
that can nevertheless be made about the joint free cumulants of ${u_t}$
and ${u_t^{*}}$. This is an ongoing joint work with Nizar Demni and
Mathieu Guay-Paquet.

Novak,
Jonathan Massachusetts Institute of Technology

Asymptotics of Looped Cumulant Lattices

Since a noncommutative probability space is an algebra together
with one of its characters, the character description problem (given
${A}$, describe ${Char(A)}$) and character evaluation problem (given
${a \in A}$ and ${\tau \in Char(A)}$, compute ${\tau(a)}$) figure largely
in the theory. While explicit solutions of these problems are classically
known for many finite-dimensional or almost finite-dimensional algebras,
free probability asks us to solve these problems for free algebras.
I will describe the formalism of "looped cumulant lattices",
which produces special characters of free algebras that are computable
in terms of noncommutative differential operators. The name comes from
the fact that LCLs are often realizable as cumulants of polynomials
in random matrices.
This is based on ongoing work with Alice Guionnet.

Nowak,
Maciej
Jagiellonian University, Cracow

Spectral shock waves in dynamical random matrix theories

We obtain several classes of non-linear partial differential
equations for various random matrix ensembles undergoing Brownian type
of random walk. These equations for spectral flow of eigenvalues as
a function of dynamical parameter ("time") are exact for any
finite size N of the random matrix ensemble and resemble viscid Burgers-like
equations known in the theory of turbulence. In the limit of infinite
size of the matrix, these equations reduce to complex inviscid Burgers
equations, proposed originally by Voiculescu in the context of free
processes. We identify spectral shock waves for these equations in the
limit of the infinite size of the matrix, and then we solve exact, finite
N nonlinear equations in the vicinity of the shocks, obtaining in this
way universal, microscopic scalings equivalent to Airy, Bessel and corresponding
cuspoids (Pearcey, Bessoid) kernels.

Emily
Redelmeier Université Paris-Sud XI

Quaternionic
Second-Order FreenessSecond-order freeness was created to extend free probabilistic approaches
to random matrices from their moments
to their global fluctuations. However, the natural definition for complex
random matrices, for which any independent,
unitarily invariant distributions are asymptotically second-order free,
is no longer natural for real random matrices, which satisfy a different
definition, reflecting differences in the graphical calculuses for these
matrices. We discuss the natural definition for quaternionic random matrices.

Shlyakhtenko,
Dimitri University of California, Los Angeles

No atoms
in spectral measures of polynomials of free semicircular variables

In a joint work with P. Skoufranis, we show that if ${(X_j:1
\leq j \leq n)}$ are free variables with non-atomic spectral measures,
then ${Y=p(X_1,\dots,X_n)}$ has a non-atomic spectral measure, for any
non-constant (matricial) polynomial ${p}$ in ${n}$ variables. The result
involves an extension of the Atiyah conjecture to products of certain
algebras.

Roland
Speicher
Saarland University

Selfadjoint polynomials in asymptotically free random matrices

I will describe our recent progress on determining the asymptotic
eigenvalue distribution of polynomials in asymptotically free random
matrices. In the language of free probability this is the same as determining
the distribution of a polynomial in free variables. This problem was
solved in recent joint work with Tobias Mai and Serban Belinschi, by
further developing the analytic theory of operator-valued additive free
convolution and combining this with Anderson's version of the linearization
trick.I will only address the case of selfadjoint polynomials. More
details and also extensions to the non-selfadjoint case will be presented
in the talk of Belinschi.

We describe block modifications of $nd \times nd$ Wishart
matrices: For a self-adjoint linear map $\varphi : M_n (\mathbb{C})
\to M_n
(\mathbb{C})$, we consider the random matrix $W^{\varphi} := (id_d \otimes
\varphi)(W)$. We are interested in the asymptotic eigenvalue distribution
$\mu^{\varphi}$ as $d \to \infty$. The distributions of Wishart ensembles
were computed by Marchenko and Pastur. Voiculescus Free Probability
theory led to a new conceptual look of such laws, which can be seen
as the free Poisson distributions. The partial transpose ($\varphi(A)
= A^t$) was studied by Aubrun.
Later, Banica and Nechita recognized $\mu^{\varphi}$ as free compound
Poisson laws for a larger class of maps. In this talk we present an
operator-valued free probabilistic approach. It turns out that \mu^{\varphi}
is exactly the matrix-valued free multiplicative convolution of a deterministic
matrix and a random diagonal matrix. We use the analytic subordination
approach to operator valued free multiplicative convolution to obtain
\mu^{\varphi} numerically, for any self adjoint map {\varphi}. This
talk is based in two joint works: with Belinschi, Speicher and Treilhard,
and with Arizmendi and Nechita.

Voiculescu,
Dan-Virgil
UC Berkeley

Free probability with left and right variables

I will describe the extension of free probability to systems
with left and right variables, based on a notion of bi-freeness.

Wang,
Jiun-ChauUniversity of Saskatchewan

Conservative Markov operators from 1-D free harmonic analysis

The reciprocal Cauchy transform of a distribution has played
an important role in the scalar-valued free probability, especially
in the calculation of free convolution of measures. In this talk, we
will discuss how methods and techniques of one-dimensional free harmonic
analysis lead to a new class of conservative dynamical systems, in which
the underlying state space is an infinite measure space. This extends
the old results of Aaronson on the ergodic theory for inner functions
on the complex upper half-plane.